NARRATOR: During a trot, do horses ever have all four hooves
off the ground?

More than a century ago, a
wealthy Northern California businessman, Leland Stanford,
enjoyed running horses on his private track. After an argument
with a skeptical friend, he placed a bet claiming that there
are times during the trot that horses have no hooves touching
the ground.

To settle the wager, Stanford
hired Eadweard Muybridge, a San Francisco photographer. Muybridge
rigged a series of still cameras in a line, so that the shutters
triggered in sequence as the horse ran past. Stanford won
the bet.

Muybridge continued this work
and in 1887, he published a ten volume set of animal and human
motion studies, called "Animal Locomotion."

Muybridge's photographs opened
up a whole new branch of scientific study... If they were
the focus of a situation where students were invited to build
important ideas about motion and change, would they also trigger
the need for calculus?

CHUCK WALTER: Looks like every
little red light in the room is on, so I suppose that means
that we're ready to go.

NARRATOR: In July 1999, before
their senior year in high school, 18 students attended a two-week
Summer Institute at Brearley High School in Kenilworth, New
Jersey.

At some point in their education,
all of these students have been involved in the Rutgers long-term
study - some going as far back as first grade.

A few were part of the Kenilworth
focus group. Others traveled to the Institute from New Brunswick
and other districts.

NARRATOR: A few years before
the Institute, two mathematicians at Brigham Young University,
Bob Speiser and Chuck Walter, were invited to begin discussions
about teaching calculus to all entering freshmen. What would
the course look like? What activities would it include? How
could it be relevant -- not just to math or science majors
-- but for all students?

CHUCK WALTER: So, let me pass
you or ask people to pass you a page which has some photos
on it.

NARRATOR: One activity they
looked at was the "catwalk", based on Eadweard Muybridge's
photographs of a cat.

CHUCK WALTER: Just look at
this for a little bit if you will. These are 24 pictures,
24 frames of a cat in motion. Do any of you have cats, by
the way?

NARRATOR: Photographs are the
only evidence that exist for a real-life event - a cat's walk
across a photographer's set that took place more than 100
years ago. The photographic evidence is not clear-cut. Open
to many different interpretations, the cat problem itself
might be more typical of the real-world problems that the
students will soon solve in scientific or business careers.
The risk for the researchers was that the students might fail.
On the other hand, success could help make calculus accessible
to more students.

CHUCK WALTER: ...there's a
backdrop to the cat, kind of a grid. The dimensions of that
grid, each line is five centimeters apart. You see that indeed
these photos were taken at an interval of 0.031 seconds a
piece. Angela, what was your remark about the time frame?

ANGELA: 0.71 seconds, so it's
not that long a time.

CHUCK WALTER: Not even a second.
Let's think about the motion of this cat and in some sense,
how we might begin to describe it. We have a particular set
of questions that we would like to use to start this discussion,
and then we can like to begin to think about the cat, to try
to get a sense of how you're starting to think about it.

NARRATOR: The problem the students
had to answer was: How fast is the cat moving in Frame 10?
and, How fast is the cat moving in Frame 20?

ROBERT SPEISER: It was the
first time with high-school students. And it was also the
first time we'd ever attempted it with students who hadn't
studied calculus yet. For us this was a complete adventure
- what would happen? It was wide open, and in fact we were
very nervous about it. Suppose the whole thing just fell on
its face, suppose they got nothing? Suppose they got frustrated?
Suppose they got the feeling that this was beyond them? Suppose
we didn't learn anything?

CHUCK WALTER: One of the very
interesting things in this sequence of photos is in Frame
10, there's a huge discrepancy in the average velocities of
the cat on one side of Frame 10 and on the other side of Frame
10. so, the question comes up - Just what can we say about
the velocity of the cat at Frame 10, and how would we begin
to think about this?

ANKUR: How many centimeters
did you say you moved? Like, three?

JEFF: I said two.

NARRATOR: The students' work
was documented by four video cameras and more than a dozen
researchers.

NARRATOR: The materials the
students had to work with were the photographs, overhead transparencies,
rulers, pens, paper, and a graphing calculator.

MAGDA: I wonder if we just
measure how much distance he covered from this picture to
this picture and divide it by .031, right?

ROMINA: The cat's going different
speeds every picture.

MAGDA: Is he really?

ROMINA: Magda!

JEFF: You would think it would
be that easy.

NARRATOR: Finding the average
velocity over a given interval of time is something many students
are familiar with from middle school or even earlier. These
students, all of whom have completed at least two years of
high school math, might think the question is very simple:
Velocity equals Distance divided by Time.

However, Bob and Chuck have
challenged the students with a much more difficult question:
How fast might the cat be moving in Frame 10, at that single
moment?

Measurements made by the students
will show that the cat was going about three times as fast
between Frames 10 and 11 as it was between Frames 9 and 10.
Will the students be able to say anything for sure about what
happened in between?

JEFF: Within the first 10 minutes,
everyone has an answer. And in the first 20 minutes, everyone
scrapped their answer and is totally lost. And I mean, this
was a classic Rutgers kind of situation: We went in; we had
the two main components in finding out the speed of velocity,
and we went in and we all scribbled it down and we were like,
all right, that's it, what's next? And then we start talking
and then one of the people would bring something up and we'd
be like, "Oh, we didn't think about that." And then as you
start looking into it, you've kind of got to push what you
did away and you start over and you get into some different
things and see where it would take us.

MILIN: I was thinking we should
also do it from 10 to 11.

MICHELLE: That's going from
9 to 10.

MATT: That changes things.
That changes a lot of things.

ROBERT SPEISER: One of the
things we were learning in our experimental teaching in the
calculus course at the university was that a lot of care and
time needed to be invested in what looked like very basic
activities: making the actual measurements, going from those
measurements to calculations of average velocities over the
intervals from one frame to the next. That very basic work
had to be done, and discussed, and presented and worked through.

MAGDA: This is what we did,
we measured each box, and it's like .36 cm and it equals 5
cm., right?

CHUCK WALTER: You physically
measured each box on the page?

MAGDA: Basically, no, I measured
from this line to here. And it was like 3.6.

CHUCK WALTER: Oh I see. So
you measured it though?

MAGDA: Yeah, and divided it
by ten. I got the average.

CHUCK WALTER: OK. I see what
you're saying.

MAGDA: Each box is .36 cm.
But in actuality it's 5 cm. You know what I'm saying?

MATT: I was thinking we should
also do it from 10 to 11 because it's kind of moving a little
bit faster in 10 than it is in 9...

ROBERT SPEISER: Somewhere in
the elementary school experience, questions about speed, distance,
and time begin to come up. A car is going at 45 miles an hour:
How long does it take for it to go a certain number of miles?
Kids need opportunities to think about that kind of problem,
because I think it gets very deep into what multiplication
and division get at. So, right there, the seeds are being
planted for this later, much more detailed study of motion,
in which now we're going to look at changes of speed.

MICHAEL: ...That's what it
says. From here to here, it's going four centimeters a second.
Does it make sense to you? It doesn't make sense to me. But
right there, he's moving four centimeters a second. It doesn't
make sense because from here to here he moved two.

__: Right. Centimeters.

MICHAEL: Oh, he's moving two,
and that small amount of time, he'd have to be moving 64 centimeters
a second, so it doesn't make sense.

__: Isn't that an awful lot?

MICHAEL: Well, if you think
it moved two centimeters, which is not that big, but in .031
seconds, which is a very small amount of time, and 64 is what
- like this big? But in a second? It just doesn't fit.

ROMINA: Well, what we did is
we went frame by frame.

MAGDA: That's a good idea.

AQUISHA: ...(inaudible) right
over the same one. Look.

AQUISHA: At my table, there
is about four kids that have been like best friends - and
there are six of us - since first grade. And that was a little
intimidating to me and I sat there really quietly, just trying
to figure them out and trying to get in the conversation somehow.
And the first day, I was extremely quiet, but after a while,
they made me feel a lot more comfortable. And I think it's
easy to work with them because I find that they're open-minded
and if you have something to say, they listen. We listen to
each other and we work things out.

ROMINA: Okay, we worked up
to 10 and then we just trash our idea.

AQUISHA: I wish these were
different colors though... We can color in all the cats like
green.

ROMINA: Yeah, why don't we
- we could do it on the overhead. So we could see it better.

AQUISHA: But I mean like, when
we put them on top of each other, they're both black, so it's
hard to even tell. I'm going to try that. I'm going to color
them all in like red or something.

NARRATOR: One of the challenges
the students faced was finding a way to take measurements
on the photographs. Aquisha invented a method of lining up
each frame with the previous frame and marking the distance
directly. Next, they organized these measurements into data
tables.

ROMINA: It is in centimeters,
but -

MAGDA: We have two, so .1 centimeters
- you've got to change it into the actual measurement.

ROMINA: Yeah, this is like
actual. It is actually what it is on our paper.

AQUISHA: Oh that's what I wasn't
getting...

ROMINA: Yes, so it can't equal
5 cm. because it's not in proportion. We haven't put it in
proportion yet.

MAGDA: So we've gotta put it
in proportion an then the speed will be right...

ROBERT SPEISER: For us, a representation
is a way of showing information. There's a tendency to think
of math problems as things that are done wordlessly in the
hand or wordlessly on paper with just figures or symbols.
But even thinking privately, we're using the paper as a kind
of memory device or maybe more than that, an actual medium
for thinking. Then when we work publicly, we're trying to
explain why we think that something is true. How can we bring
evidence forward without using a representation of some kind?
I don't know how we can think without representation.

NARRATOR: If representations
are the currency of mathematical discourse, then this activity
would be rich in different ways of looking at the problem.

JEFF: ...How many things over
is it?

MICHELLE: Ten.

JEFF: It's 10 cm from the first
to the end.

MICHELLE: 10 cm. two things.

JEFF: From here to here?

JEFF: Which line is this? Is
this from the end of the 10 or the end of the 5?

MICHELLE: That one is the end
of the 10.

JEFF: So he moved about maybe
2 cm from 9 to 10. If that's the case, dude, he's moving ridiculously
slow. If we just show from 9 to 10. You know? I mean, he's
like barely creeping.

MILIN: Yeah, but you have to
divide by .031, so that number will get a lot bigger.

MICHELLE: Right now my group
has three different answers. We don't know if our answer is
right. We don't know if the way we're doing it is the average
velocity from the beginning to the time frame or if it's the
velocity at that instant at that time frame, and that seems
to be the major problem right now.

ROBERT SPEISER: If you want
to figure out what's going on in Frame 10, you have some choices.
For example, should you work out the average speed from Frame
1 through Frame 10 and then the average speed from Frame 10,
say, to Frame 20 and then compare those? Or should you look
just locally, right near Frame 10 - at what happens from Frame
9 to Frame 10 and then from Frame 10 to Frame 11? Why would
you prefer one over the other? Gradually in their arguments,
they zeroed in on the local information around Frame 10. I
think that was a huge step forward.

CHUCK WALTER: Okay, can we
let this group come to the board up there and tell us what
their thinking is?

NARRATOR: At the end of the
first day, Chuck asked the groups to summarize their progress.
Starting from the location of the cat's nose in Frame 1, Michael's
group made this graph of the distance that the cat traveled,
plotted against time.

MICHAEL: Well with this graph,
it's not the velocity, it's just the change in distance. But
what I see here is a certain velocity and another one. So,
the time would be as you go along, you know, and the distance
is the difference between each and every one. So, that means
he's walking at a constant rate for a very small amount of
time.

Then when this changes, that's
around when he starts crouching down and he starts walking
- and 10 is around here and he starts going at this speed.
This is not accelerating here. He's actually going the same
speed. That's why all these guys are like the same apart -
thatís basically a line. It shows two lines.

It may be a little off in there,
but basically what it is - he's walking in the beginning going
this speed and running at the end, going a faster speed.

MICHAEL: In a traditional way,
a teacher would bring out an idea or like something like that,
but in the Rutgers way, a kid would come up with something
and show it to the class. With the cat thing we're learning,
either me or Matt or Romina's group would just go up there
and we would tell each other - "Well, this is my graph and
this is what it shows. I'm teaching you about what I discover,
you know?" And like, that wouldn't happen in a regular class.
By us going up there and showing us what we got, it helps
other groups use that information to finish their task or
whatever.

[MUSIC]

NARRATOR: The next day, the
researchers asked the students to continue their presentations.

ANGELA: Okay, what we did is
we did the average. We measured the velocity between the 9th
and the 10th frame and then the 10th and the 11th frame. So,
we figured that if we took the velocity between these two
frames and these two frames, and then figured out the average,
it would give you like the middle, which is the 10th frame.

NARRATOR: Angela & Shelly's
group came up with precise numbers for the velocities in Frame
10 and Frame 20.

ANGELA: And we got 145.161
centimeters per second for the velocity. Then we did the same
thing with the 19th and the 20th and then the 20th and the
21st and we got 354.839 centimeters per second.

NARRATOR: Romina transferred
the data from her handwritten table to her graphing calculator.
This allowed her to share two more representations with the
class.

ROMINA: This is our time versus
distance. At first it's going slower 'cause it's walking and
then it accelerates. It's going more distance because it's
like running, galloping, whatever. And then we also graphed
our distance versus speed, and it's a little different from
Mike's because it shows, it shows like the cat... We kind
of agree with Mike at the first part where it's walking, it's
going at like a constant rate. But then it shows when it starts
like its gallop, it kind of like pauses a little bit and then
it starts accelerating faster and faster until it reaches
like a climax and then it starts coming down. So, towards
the end, that's where the cat's stopping. Which Mike's didn't
show that; it kind of just showed like a constant rate at
walking and running. So, ours just kind of disagrees with
theirs a little. That's all we did today. Aquisha, you want
to do your thing?

NARRATOR: Building on work
from the previous day, Aquisha came forward with a powerful
representation: one that changed the way the entire group
approached the problem from this point on.

AQUISHA: This is what I did
yesterday. By putting the two of these on top of each other,
like Frame 1 over Frame 2, you see the distance that the cat
traveled. I measured from the nose here to the nose of the
next one. I used the lines, the thick lines, to line them
up.

NARRATOR: Building on work
from the previous day, Aquisha developed a unique way of representing
the movement of the cat. Alternating two colors of marker,
she traced the distances the cat ran between frames as short
line segments lined up end to end.

AQUISHA: This represents, from
frame to frame, what the cat traveled. Like this one's from
Frame 1 to 2 and then from 2 to 3, up to 24. And today, I
put them all together. This is just all the distances. I connected
them.

NARRATOR: Aquisha's line representation
clearly shows the central issue of this problem: there is
a big change in the cat's velocity, right around Frame 10.

ROBERT SPEISER: The representations
that Aquisha is showing us were entirely her own. She's building
from the same numerical database that Romina and Magda are
working with. But she's representing the information the numerical
information in that database in a very strikingly different
way, in a much more visual way. A much more personal way.

CHUCK WALTER: Thatís
a remarkable idea. What kind of sense of the cat's motion
can you see here? Can you see anything at all? What do you
think?

BENNY: I guess you see the
total distance traveled by the cat.

AQUISHA: You can see the difference
that the cat was walking slow and how it got faster.

JEFF: You can see how, kind
of like Mike was arguing the other day, how he's going the
same speed a lot in the beginning, you see a little time where
he's accelerating, and then he pretty much runs along the
same at the end.

CHUCK WALTER: Does that picture
that Aquisha's presented there carry the same sense that some
of the other graphs that we've seen does? Could you say why,
how?

BENNY: I think Jeff said you
can see how in some areas the speed stays at a constant rate,
then changes slightly, then as it changes it stays at a constant
rate. You can kind of see that from that.

CHUCK WALTER: How would you
see that, Benny? Tell me a little more about what you see...

ROBERT SPEISER: Students find
the standard scientific representations such as tables and
graphs a lot more difficult than we often think that they
do. It seems that unless they have the opportunity to build
personally meaningful representations, whether they're standard
ones or not, and work through what the information is that
is carried by this representation and how to read it, then
the really important standard scientific representations,
like tables and especially graphical representations, that
are so important remain difficult, remain problematic for
them. Aquisha's, I think, is a really beautiful example of
how just wonderfully creative and powerful a personally-developed
representation could be.

AQUISHA: The first dot represents
from one to two.

BENNY: Okay. So, I mean five.
So, after five, it begins to accelerate.

CHUCK WALTER: Thanks, Benny.

ROBERT SPEISER: In a sense,
Aquisha kicked open the door, because we need to understand
how the cat was moving in space, and laying it out along a
line is the first step to seeing that. It was delightful to
see this, partly because of the visual power of this simple,
modest looking image, and the effect it seemed to be having
on the other students. Then the next step, of course, was
to encourage them to go further, to build this in space, you
know, in a much larger frame, in a frame that they could physically
move through...

CHUCK WALTER: I've got an idea.
Be a cat, okay? I'm wondering if, if in fact we can do this,
if in fact we might be able to be this cat. If we think about
this, this is how the cat moved, right? We see not only how
far it went, but we see kind of spacings of distances. Isn't
that what this represents, Aquisha?

What if we, instead of marking
this out here like this, what if we marked this out on say
the floor? You know, maybe expanded it so that it would be
not just nine centimeters long or 17 or whatever. We would
make it, make it big enough that we could put it out here.

ROBERT SPEISER: Mathematical
knowledge grows from our actual experience of moving around
and living and thinking about how we operate in the world.
What does it mean to get at the story of what the cat did?
How would you understand it? Would you describe it for me
outside, in that we have these numbers, that these are our
measurements, these are the speeds, that it fits into this
graph? That's one way to do it. That leads you to one set
of representations. Another way to do it would be to try to
do a movement that would be like that. What would it mean
to do that? How would you get at the story? That leads to
another set of representations.

CHUCK WALTER: We could mark
the frames, right? Like Frame 1 here, Frame 2 here, just like
Aquisha did. And then maybe we could see if we could move
like the cat. Is that a dumb idea or do you want to try it?

STUDENTS: Let's try it.

NARRATOR: After some discussion,
the students selected a scale of 10 times the distances the
cat ran. The runner's task was to pass each mark exactly on
the beat of a steady drum, in effect, to "be" the cat, but
on a human size scale.

[Drum beating]

CHUCK WALTER: All right, Benny
wants to go. Ashley, hang on to what you were thinking about
when you went through 20, okay? [Drum beating]

ROMINA: The actual size of
the paper layout was 130 centimeters. Then we made it a little
bit bigger in the library, but you couldn't feel the full
effect because like at first, in the span of two feet was
the first ten intervals, so you couldn't really do it. So,
we figured we'd multiply it by 50, that way we'd actually
have to be walking - from a piece, we marked the piece of
tape - from like tape to tape, we'd actually have to be walking
and then running.

NARRATOR: On Day 3 of this
activity, the students chose to work on a still-larger scale,
one that is fifty times the distance the cat actually ran.

ROMINA: Should I write-

MAGDA: You can write the distance
in Frame 1.

ROMINA: Should I write Frame
1? Like, I'll put one here and then two?

MAGDA: Yeah.

MAGDA: 375, Romina.

ROMINA: Isn't there a spot
here where it's like the same thing over and over?

MAGDA: Yeah - one, two, three,
four, five.

ROMINA: 375 is the one?

MAGDA: Yeah, 5 times 375.

BENNY: This is basically going
to increase our stride so it makes it look like we're picking
up speed more, you know, instead of just walking like at a
steady pace, so we can see it more. Like more visibly we can
see it, instead of just walking and it looks like we're just
taking a step. See, with this, we're taking longer strides
and we can actually see it. So, that's basically it.

[Clapping]

MICHAEL: It looked like they
were going slow in the beginning, and they stopped - like
slowed down real quick, you know, when the things get really
close together, and then they start to pick up speed until
they start to run. I think they felt like at the end like
they weren't speeding up - they were just running - at a constant
speed, and it was easier to see exactly what those dots on
the graph really meant.

ROBERT SPEISER: How do those
scientific representations come to life? Well, it seems it's
pretty obvious when you think about it, although our educational
culture makes it look so foreign: get in there and move! And
then connect that with the numbers, with the graphs, and with
the other pictures, and let's see what kind of sense we can
make through connecting all these things.

ROMINA: It made sense of it.
Like, we had the numbers there, but they didn't make sense
to us. We didn't understand how something could change speeds
so fast, what was going on, why it would change speeds so
fast. But when you do it like a real life version of it, you
can see what the cat's doing so you understand it. And like
our graphs would go up all of a sudden and like fly down.
But then when we did it, we could see like it was accelerating,
it came to a peak, and then it was slowing down. It just made
sense of all the math.

[Clapping]

NARRATOR: How might the personal
experience of running help these students to deepen, organize,
or clarify their growing understanding of the motion of the
cat?

PART 2. "BETTING
ON WHAT YOU KNOW"

NARRATOR: After running the
model in the hallway, the students began an open discussion
about the cat's motion.

CHUCK WALTER: I was wondering
what kinds of questions or what kinds of responses came out
of this morning's little workout there. Let's be kind of brutally
honest with each other and bring real questions out on the
table and then we can address those. Jeff?

JEFF: On the brutally honest
bit, I'm just a little confused. I thought it was good to
make a model. I think that's always good to do something,
like if you can do it physically yourself, but I don't see
what we're getting from doing all of it.

CHUCK WALTER: That's a pretty
brutal question, right?

JEFF: Yeah.

CHUCK WALTER: But I think it's
also right to the heart of the matter and the point. Okay?

ROBERT SPEISER: I think Jeff's
question is a good question. Some work still needs to be done
to connect what was done in the hallway with the graphs and
the tables and the other representations, including Aquisha's,
that had been developed before.

MATT: I just think that we
finally figured out, like what Mike said, how he thought the
cat stays the same speed or whatever. The cat doesn't stay
the same speed. It's constantly speeding up and speeding up
and speeding up.

VICTOR: And the way I see it
is I think that the graph, whatever, that this group right
here came up with - Romina - how they showed the change in
speed and position in time and all that stuff. I think whoever
ran, they can kind of feel it, because in the beginning, they
were running at a constant pace. Then they kind of slowed
down; then they picked it up, picked it up and kept on picking
it up, which was what they showed.

CHUCK WALTER: So, what you're
saying is kind of adding to what -

VICTOR: - He said, but also
proving their point with their graph.

CHUCK WALTER: That was an interesting
graph. I'd like to take a look at it again sometime if you
guys still have it.

ROMINA: I'll check. Hold on.

CHUCK WALTER: Anything else?
I want to hear what Benny says because Benny's been a real
spokesperson for a lot of this over this period of time.

BENNY: Well, I was just saying
how this model and with the model outside, I was thinking
it gives us more of a visual explanation and not like a mathematical
explanation. You know, like you can actually see like between
point 10 and 11; this is where it picked up in between points
19 and 20; this is where the speed begins to get fast. Just
something you can see: All eyes, all eyes on the cat.

ROBERT SPEISER: It seems to
me that Benny is trying to bring the discussion very strongly
back to what is the cat actually doing. I think the "seeing"
is a little deeper than literal, physical seeing. He's going
for the abstraction that's behind the different representations,
that no one representation captures the idea of what's happening,
but somehow we need many of them.

NARRATOR: Romina re-displayed
her graph of Velocity over Time.

MICHAEL: Your time was - how
does it go up, from .031 to .062?

ROMINA: Yeah.

CHUCK WALTER: And is that time?
It looks like time across the bottom.

ROMINA: The x is time and the
y is velocity.

MATT: Each of these little
dots here, these are all his velocities at a certain time.
So, if you see how the line goes, that's his climb in velocity.
Yeah, this is like his acceleration. You can tell, like from
here to here, his acceleration goes down; and from here to
here, it starts to skyrocket up like that. Then he evens out
for a while; then it goes down a little bit; then it skyrockets
up again.

CHUCK WALTER: Where is the
acceleration?

MATT: These lines. This line
here, this line here... Where it starts to swing up.

ROBERT SPEISER: Matt had a
graph of velocity or speed plotted against time. He was referring
to it as an acceleration graph. So, Chuck's question is inviting
him to explain where he saw the acceleration, and then he
does it splendidly, by looking at where the velocity was changing
the most rapidly.

MAGDA: Where is he going at
a constant speed?

MATT: At these ones here, he's
going at a constant speed.

ROMINA: The photographer couldn't
tell the cat, "Okay, you're going to start walking; then just
speed up into a run." So, the cat was probably walking and
then he did something to like it make it, like scare it. And
then all of a sudden, it went from like a constant walk, just
like regular walking, to like speeding up into running, because
something had scared it and made it move.

VICTOR: Right, right, right.
What I see, right, is like these constant speeds, it's like
the first constant speed, what I figure is that he's most
likely to just walk. But then at the second constant speed,
I think, you know how he just shoots up for a quick minute
at that first "A" that Matt has drawn up? I think that's when
he jumps up. Then while he's in the air for that like split
frame right there, he's in the air traveling and that's constant,
and that's when he kind of comes down, like after the constant
speed that he separated, and then he just starts walking again.
That's what I gathered, I think he just stops because he doesn't
keep going.

MATT: I'm not even sure - how
many steps does that cat take? Because this here, this might
be when he pushes off and what speed he gets to. and when
he pushes off. And then this would appear to be what speed
he gets to, and then he would push off again. That's what
I'm thinking too.

ALICE ALSTON: Oh, what, though,
you were looking at our pictures?

BENNY: Yes.

ALICE: How many steps did you
think the cat took?

BENNY: Like four or five, maybe
four or five. Because at the end there - Somebody got the
thing?

MATT: That's like four or five
because -

BENNY: Yeah, that's like four
or five. Now if you look at the first step, he begins his
step right here. And if you look, he's lifting his leg up
gradually, gradually, gradually, gradually. He begins to put
his foot down here and then his foot touches the base right
here. That's one step. Now, he begins another stride right
here, gradually lifts his back leg up. You can see - this
is the foot we are going to judge it by - his back leg, when
he lifts his back leg up again and he lifts it up high, high,
high, then he begins to bring it down, touches base again
- that's the second step. Then as his speed picks up, it starts
to take him less time to take a step. So, in this third one,
his back leg - touches base here and then he begins another
stride right here.

MATT: Basically he took like
four steps.

BENNY: Four steps.

MATT: When he's walking, you
don't see that acceleration; like, you're not going to see
it on this graph. Like, this is his first two steps here,
while he's walking at a constant speed. And all of a sudden,
on that third step, he gains his speed, and this is when he
lands, and then that's his speed again, when he pushes off.

CHUCK WALTER: This is really
getting interesting. It looks to me like there are some different
kinds of things entering the discussion. First of all, we
have - you produced some data; you produced some graphs, lots
of different kinds of graphs. But there's also I think in
the mix now, the ideas that are starting to come out of our
walking/running cat experience. And of course, there's still
the photographs.

ROBERT SPEISER: Romina began
this series of connections between the cat's actual movement
and the velocity graph. Several of the other students, culminating
with Benny, developed that idea. So it's the first time that
the idea of energy has come into the mathematical discussion.
Where was the force applied? You can see that in Muybridge's
photographs.

NARRATOR: The next morning,
Day 4, the researchers posed a hypothetical problem, with
the students' math teacher, Ralph Pantozzi, pretending to
be the CEO of a large corporation.

CAROLYN MAHER: Let me say a
few words. We have a 10:30 deadline. But I'm a little concerned....

ROBERT SPEISER: Getting down
to the bottom line - whatever you say, how you justify it?
For the last 10 years, these students have been part of a
study of justification, explanation, and proof. And it's an
invitation for the whole community to fit some reasoning together,
and to present it.

CAROLYN MAHER: ...But I'm really
concerned because my job is on the line. Mr. Pantozzi wants
a report at 10:30 and he put me in charge of this project.
You're all the wonderful consultant staff that has been working
to help me on this. So, I'd like to hear- I mean, I don't
know what's going on. I've been called from meeting to meeting;
I've been really sort of distanced from what you've been doing
this morning. Why don't you tell me what the problem is exactly,
because I'm not even so sure I understand what the issue is
here.

VICTOR: He's making his wager
on what the speed is at Frame 10 and what the speed is at
the 20th frame as well. But after we did our calculations
-

CAROLYN MAHER: Hold on a minute,
Victor. The exact speed? How accurate?

MATT: Yeah, but if he wants
results, he's got to specify what he wants. We can't -

CAROLYN MAHER: Well, he's going
to make this wager or not make this wager, and he's- How right
does he have to be? We have to advise him?

VICTOR: Now, when you listen
to our statement, then you will understand. We wrote, we believe
that this would be the best position to take on the cat's
motion. However, we believe that this is an approximation
rather than an exact answer. We will not be held responsible
for any monetary losses. [laughter]

NARRATOR: Each group came forward
to present their findings. Victor's group presented the same
velocities for Frames 10 and 20 that they had shared on Day
2 of the investigation.

VICTOR: Okay, now, what we
believe is that you can actually tell the velocity of the
cat at Frame 10 and 20, but you don't necessarily know how
accurate our numbers are because everything we had to do,
we kind of like eyeballed our measurements and this is not
a real life scale. So, we can't really get as accurate as
we want. What we did was, we used the basic physics formula
of 1/2 times the quantity of the initial velocity plus the
final velocity, and we have a velocity of about 145.161 centimeters
per second and give or take 20 centimeters/second due to inaccurate
measuring. And for Frame 20, we have about 354.839 centimeters
per second. If you look at the picture, you see that about
Frame 20, he's really taking off at almost twice the speed
because he's springing forward from the gallop. So, you can
see actually the change in speed being that drastic from Frame
10 to 20, so that's why our numbers, you know. But, we do
not suggest that he make the wager, however, because gambling
is illegally and morally wrong. So there you go. Any questions
anybody?

ROMINA: Yeah, I've got a question.

VICTOR: I thought you would.

ROMINA: When you say you take
the average, right, you're saying you start off Frame 10 with
the 2 - it's not the velocity - whatever that 2 divided by
.031 is, right? And you end Frame 10 with 7 divided by .031.
So, that's a really big jump. How do you - when you take the
average, where are you saying? Is that like right in the middle
of Frame 10, that's the speed it's going?

VICTOR: Right, right here?
Exactly. Because from Frame 9 and 10, it's moving - the change
from 9 to 10 is about 2 centimeters and from 10 to 11, which
is the next frame over, it's 7 centimeters. That's just because
of what the movement of the cat did.

ROMINA: So, in Frame 10, though,
he's going like -

VICTOR: Halfway, something
like-

ROMINA: But in Frame 10, he's
changing speed, so he's going, he starts - In Frame 10, he
has to somewhere - up from like 80 to like - our measurement
said 80 centimeters per second to like 200 centimeters per
second, when we divided our numbers. So, that's like a big
gap. Like, you can't - Why did you take an average on that?
You don't know exactly where he's taking the - Like, I can
understand in the second one, the average is a little bit
more like realistic; like you could bet on that one. But the
first one - you can't bet on that number because he's going
so many different speeds in that one frame. You understand
what I'm saying?

MATT: Yeah, we can't bet on
it.

ROMINA: Yeah, exactly.

VICTOR: That's why we try to
take an average, you know, a guessed average. I mean, that's
why we said that it wasn't really, really accurate, especially
since the frame's before that, he was like doing all these
other things. But the second one, we can't bet on it.

MATT: We wouldn't recommend
he bet on it. I wouldn't bet on it.

ROMINA: Thank you.

VICTOR: No more questions?
Okay then. I guess we're done.

TIM SWEETMAN: It sounded to
me like your recommendation was don't bet, right?

VICTOR: Right, don't bet. I
mean, if you had to be real exact, I'd tell him not to wager.

JEFF: If it just had to be
to the second, would you tell him to bet?

VICTOR: If they had to be to
the second-

JEFF: You think he's going
to land in there around three or whatever the number is?

VICTOR: No, not really. When
you have to be exact, I don't think you should bet on it.

NARRATOR: Shirly and Mike's
group presented next.

SHERLY: Okay, what we did was
- Because originally, we measured from the cat's nose or whatever,
but we took a point in the body that doesn't really move that
much, so we took the point for the nose and the point for
the base of the tail and we averaged it together for Frames
9, 10, and 11. And then he found the difference between the
distances and he got the velocity by dividing by the time
and he got those numbers and he averaged those two together
and he got the average speed, basically for Frames 10 and
20. But they're not accurate, so we don't recommend him betting.

MICHAEL: Don't bet.

TIM SWEETMAN: That's it? Don't
bet? Well, we already bet.

MICHAEL: Since all three groups
did this and got three different answers, I'd say don't bet.
Because if we all got the same number, like 149 point something,
go for it. But if it's like 160, he got 80 or something, and
we have 140, just don't.

ROMINA: Yeah, we all had the
same process, but like different-

TIM SWEETMAN: How could you
have the same process and get 80?

ROMINA: Because our measurements
are a lot different than theirs. And we eyeballed it on and
they used a ruler.

MICHAEL: I don't know what
to make of this, but I would say don't bet.

TIM SWEETMAN: All right, let's
hear what this group has to say. I want to see where this
80 came from. It bothers me a lot that we're so far off. He
won't be happy with me when I tell him this.

ROMINA: What we did is we started
off with like measuring like with our ruler and like where
the cat's nose moved, but then that wasn't working for us
because the measurements were getting too small, so then we
kind of eyeballed it. And we said from 9 to 10, he moved about
half a box, so that's 2.5 centimeters. And when we divided
that by .031, it came out to 80.

TIM SWEETMAN: You used the
nose?

ROMINA: Yeah. Our numbers go
from 80 to 200; that's like a big difference. So, if you take
an average - like, we just said we wouldn't bet on an average.

ANGELA: Frame 10 is right in
the middle of those two, so it's going to be the average.

ROMINA: Yeah, but in Frame
10, he's going-

MATT: It doesn't mean the average.

ROMINA: There's a lot going
on in Frame 10. I just don't think he should bet on Frame
10. Frame 10 is too like - Frame 10, too much is going on
in Frame 10. Like, like 'cause if you're going like technical,
the guy could be like, "Well, I meant at the beginning of
Frame 10 what his speed was," or the guy could say at the
end of Frame 10.

ROBERT SPEISER: What we're
sure of is that there was a large change in the velocity.
What we can't be sure of is what that velocity might have
been just at that moment when Frame 10 was exposed. It's a
very hard thing to talk about. It's not the speed, somehow,
that we can lay our hands on around frame 10. It's the fact
that the speed is changing so much. And yet, we don't know
the speed at frame 10. And probably we can never know it.

ROMINA: Like for Frames 19
through 20, if I measured - we didn't get our measurements
right, really measured it, like 10, like 20- you could bet
on that one, because that one, we could get our measurement
accurate enough.

MATT: At Frame 20 where the
numbers are so close, I mean you have the exact - most of
us have the exact same numbers. So, if you have the exact
same numbers, the exact same distances, it's a constant speed.
So, if you can take that and take the average of that and
come up with your speed from there, that's a good number.
And if we all get the same distance for that, we can give
him that number and he can probably bet on that number. But
in Frame 10, I mean it's worthless.

TIM SWEETMAN: Questions from
anybody else in the room?

ROBERT SPEISER: I think it's
the demand for the explanation, the demand for a reason, that
pushes you through the surface features of the situation and
really gets you into the heart of it. I think that one of
the most important messages from this particular long-term
research study has been how powerful that need for justification
is, both in terms of good teaching - helping the students
to build the most powerful and lasting understanding of important
ideas - but also how central it is for the students themselves,
how important it was for them, for the mathematics and their
reasoning and their conclusions, to make sense, to be explainable,
to stand up to public discussion. Their own demands for rigor
turned out to be one of the findings of this study.

ROMINA: You're going from-
there's exit 9, 10, 11 on the parkway. This is how we like
thought about it. At exit 9, you're going 30 miles per hour.
And exit 11, you're going 60. How fast where you going at
exit 10? You don't know. You could have sped up like - you
could have been going 60 there; you could have been going,
you could have still been going 30.

JEFF: You could have sped up
to 120 miles per hour for exit, for that exit 10, and slowed
down to get 60. I mean, all you know is the beginning and
the end. You have no clue what he did in between.

JEFF: You know, since the CEO
is here now...

ROMINA: So nice of you to stop
by...

CHUCK WALTER: Oh my gosh.

RALPH PANTOZZI: I was working
on something of great concern to most of you. I need to go
back to that meeting and I want to know is should I call off
the bet? I have just a few more minutes to call off the best.

STUDENTS: Call it off.

RALPH PANTOZZI: I'd like to
be convinced of why I should call it off. I could make some
money here.

JEFF: But all of us are telling
you that we all have different answers and we're not sure
that they're right. Isn't that enough for you to call it off?

MATT: We're all taking averages;
they're all approximations. We don't have exact measurements.

ROMINA: Yeah, our measurements
are even off. Just right off the bat, we're just not doing
good.

ROBERT SPEISER: They wanted
a complete and almost total understanding of what was actually
happening in this situation. That was not something that we
posed to them. It was something they demanded for and of themselves.
And not only did they demand it, they delivered.

RALPH PANTOZZI: Well, let me
ask this then. If I call off this bet on Frame 10, I guess
I need a reason to - if I could just state a reason for me
to make a bet. Like, if the situation was such, I should make
a bet, and if the situation is not such, I should not. Could
you make a -

MATT: It depends on the motion
of the cat.

VICTOR: Right, and then you
have to ask them how accurate does he want his measurements
anyway.

MATT: If the situation is such
that the cat is moving at a constant speed, meaning that there's
the same distance between frames, then you should make your
bet. But if not, in the case of Frame 10 and Frame 9 to 10
and Frame 10 to 11, it jumps from 2 centimeters to 7 centimeters
difference. That's a big difference. You don't know what happens
in between there. So, in that case, you don't make the bet.

ROMINA: People underestimate
what we can do. And if you just give us problems and we keep
working at it, it builds us up; it just develops your idea.
And maybe and like when we're running the world we can come
up with better solutions because we know more and we can like,
we've practiced it and we've been able to like have like group
thinking and solutions.

[MUSIC- Kenilworth high school
graduation]

NARRATOR: Eleven months later,
the students who were seniors graduated from high school.

NARRATOR: What conditions
in the mathematics classroom might be required in order to
make mathematics a meaningful subject for all students, all
the way through high school?

ANKUR: Math is, I guess, part
of my daily routine. Itís been there since I could
remember, every day itís existed.

ROMINA: It's more than just
the numbers to me. It's like you have to go deeper, you have
to, if you understand something from the beginning, you're
going to always understand it. You can't forget something
like that.

BRIAN: It just helps me doing
things, other than math, too. Like I think more in depth,
very seriously about things.

MICHELLE: Now I realize, even
in my high school, when I'm doing problems, I can understand
a lot of times the steps I'm doing, and other people don't
take those steps to look at it more analytically.

VICTOR: Nobody comes up with
a solution to a problem by themselves. People have to get
together and think about a problem, and get what they know,
and share the information, and come to one exact solution.

BENNY: most people get stumped.
But you shouldn't give up. You should always try to find methods
around. You have peers. You have teachers. They're all there
to help. And I found out that you can use those people as
resources to help you.

JEFF: We were all different
teams and it was just like the real-life setup. And our group
of people had to come up with an answer. And even though we
all had different answers we had to refine it to a point where
everyone was happy with it. And you're going to come in contact
with that no matter what you're doing.

NANCY BATON: Michael John Aiello...

[Cheering]

NANCY BATON: Ankur J. Patel...

[Cheering]

NANCY BATON: Brian Malina...

[Cheering]

NANCY BATON: Magdalena S1owolwsky...
Magdalena has receives a medal for achieving the second highest
average in Advanced Placement Calculus.

[Cheering]

MICHELLE: I think that, in
the end, it just requires a teacher paying a lot more attention
to the students, and to what every student is thinking. But
I think it's worth the investment of time, because not only
does it help the students with math, but it helps the students
work through other problems in any subject, that the way you
go about the problem, and the way you look at it and think
about it can be a lot different. And it makes the student
feel more important, because you think the teacher cares about
you, and cares about what you're thinking.

NANCY BATON: I now declare
the ceremony officially over. Congratulations to the David
Brearley Class of 2000.